Processing, Modeling.pdf

Vol. 11

PROCESSING, MODELING 263

PROCESSING, MODELING

Introduction

A model is a quantitative abstraction of a physical process, in which the descrip-

tion of the process is represented by the solution to a set of mathematical equations

(1,2). The model equations represent the behavior of the real process to the extent

that the equations embody an accurate description of the underlying physical and

physicochemical phenomena. The mathematical formulation enables the model

to be used for a variety of purposes, including design, control, and exploration

of operating strategies; the effects of changes in process variables and material

properties can be inferred from the model without extensive experimentation.

Mathematical models have long been used for these purposes in the chemical and

petrochemical industries, and computer-aided design and computer-aided manu-

facturing (CAD/CAM) have been taking on growing importance in some polymer

processing operations.

The essential elements of any model of a physical process are threefold: the

geometry, the relevant laws of physical conservation (mass, momentum, and en-

ergy), and the speciﬁc constitutive relations. In polymer processing applications,

the last of these could include the stress-deformation equations for the material be-

ing processed (the material rheology) and the kinetic equations for phase change.

The mathematical complexity of a model depends on the process and the type of

information required. Qualitative information can often be obtained from a model

that greatly simpliﬁes the physical phenomena to obtain mathematical simplic-

ity and consists only of one or more algebraic equations, for example, whereas

structural predictions typically require the consideration of details of stress and

ﬂow ﬁelds and hence the solution of nonlinear partial differential or integral equa-

tions. Advances in computing technology have made the latter goal possible for

some processing applications (3,4).

The term computer model is often employed as a consequence of the exten-

sive use of computing technology in conjunction with mathematical models. This

terminology is unfortunate, because it confuses modeling and simulation. The

equations describing the physical phenomenon make up the model and incorpo-

rate the full understanding of the process. The model is independent of the means

that are used to solve the equations, which might be analytical or numerical, with

or without the use of computers. Application of the model to simulate some speciﬁc

situation requires the choice of a solution technique; numerical artifacts associ-

ated with the solution method could introduce errors that are not inherent in the

physical assumptions on which the model is based. Comparison of the predictions

of a model with experiments requires careful separation of the effects of modeling

assumptions and numerical techniques.

Polymer processing operations include polymerization reactors; devices for

mixing, conveying, and extruding molten polymer or polymer solutions; and de-

vices for forming shaped objects in the liquid state and post-forming solid-state

operations. The modeling of chemical reactors to predict conversions and prod-

uct distributions is a mature art (5). Efforts in this ﬁeld have focused on pro-

cess control and the conditions leading to instabilities and “runaway” reactions.

264 PROCESSING, MODELING

Vol. 11

Polymerization reactors form a subset of the subject of reaction engineering, dis-

tinguished only by the need to know the details of the speciﬁc reaction pathways

and kinetics, and perhaps the dependence of viscosity on conversion. Polymeriza-

tion reactions and reactors, including reaction injection molding, are treated else-

where, and this article is concerned only with the modeling of the mechanics and

structure development in mixing, conveying, extrusion, and forming processes.

Rheology

Models of the mechanics of polymer processing operations can be categorized by

the rheological description used. Low molecular weight liquids are Newtonian,

which means that the stress at any point in a deformation ﬁeld is a linear function

of the instantaneous rate of strain and is independent of any prior history; the

stress in an incompressible Newtonian liquid is characterized by a single material

parameter, the viscosity (

σ =−

p I

+ τ

(1)

τ = η

[

∇

(

∇

v ) T ]

(2)

Here,

is the total stress, p the isotropic pressure, I the identity (unit)

tensor, and

the extra stress (ie, the stress in excess of the isotropic pressure).

is the gradient differential operator, and v is the velocity vector; T denotes the

transpose of a tensor. For a one-dimensional ﬂow with a single velocity component

v , in which v varies in a single spatial direction y that is transverse to the ﬂow

direction, equation 2 simpliﬁes to the familiar form

τ = η

d v

d y

(3)

is constant over the entire ﬂow ﬁeld), substitution

of equations 1 and 2 into the balance of linear momentum leads to the Navier–

Stokes equations, which have been the subject of intense study in classical ﬂuid

mechanics for more than a century.

In contrast, the stress in a macromolecular liquid depends in a nonlinear

manner on the entire history of deformation. The nonlinearity is manifested by

phenomena such as the strain-rate dependence of the shear viscosity, strain hard-

ening in extension, and the transverse normal stresses in shear ﬂow that cause

extrudate swell. The dependence on history is demonstrated by the transient

buildup and relaxation of stresses following changes in ﬂow variables, and by the

existence of the dynamic storage modulus ( G ) in oscillatory shearing. There have

been many routes to the development of appropriate stress constitutive equations

for use in processing ﬂows of polymeric liquids (6–8), including general continuum

mechanics formulations and molecular and quasi-molecular theories. No consti-

tutive theory developed to date is adequate for quantitative predictions in all

processing ﬂows, in some cases because the complexity of the equation prevents

), which can be temperature- and pressure-dependent

but is independent of the deformation rate:

∇

When the ﬂow is isothermal and the liquid has a viscosity that can be taken to be

independent of pressure (ie,

Vol. 11

PROCESSING, MODELING 265

v ) T ].

[An invariant of a tensor is a quantity that has the same value regardless of the

coordinate system that is used. The second invariant of the deformation rate ten-

sor, often denoted II D , is a three-dimensional generalization of 2(d v /d y ) 2 , where

d v /d y is the strain rate in a one-dimensional shear ﬂow, and so the viscosity is

often taken to be a speciﬁc function–a power law, for example–of ( 2 II D ) 2 .]

The dependence of the stress on the strain-deformation history of macro-

molecular liquids can be incorporated in two ways. The stress constitutive equa-

tion can be formulated as a differential equation, in which the extra stress

∇

(

∇

the solution of an equation that is typically of the general form

A (

τ /

G )

· G + λ D

(

τ /

t = λ

G )

[

∇

(

∇

v ) T ]

(4)

/ G ) is a nonlinear tensor function of the extra stress; the simplest

such form is a Maxwell liquid, in which A is equal to the unit tensor and the

ﬁrst term in equation 4 simply becomes

is the time constant for stress

relaxation, and G is the shear modulus, which is a material property; the shear

viscosity

/ G .

t denotes a nonlinear time derivative that

accounts for the invariance of the physical quantities under changes of the frame

of reference of the observer;

is equal to the product

G .

v , but the precise

form depends on the particular constitutive theory. Equations of this type follow

from transient-network theories, which are based in part on concepts embedded in

the theory of rubber-like elasticity. Equations of this type also follow from “tube”

or “reptation” theories, in which the chain in an entangled melt or concentrated

solution is envisioned as being restricted to an imaginary tube made up of the

constraints imposed by the surrounding chains, which restrict chain motion that is

transverse to the molecular backbone. (Differential equation forms for tube models

usually require mathematical approximations that are not contained in the basic

model.) The relaxation time and shear modulus depend on the deformation rate

or stress in some theories, whereas in others they depend on a dynamical scalar

or tensor variable characterizing the structural state of the melt or solution. If the

spectrum of relaxation modes characterizing the linear viscoelastic response is to

be included, the stress is comprised of a sum of terms,

t contains terms of the form

τ ·∇

τ =

τ i

(5)

. The values of the spectral parameters

in the limit of vanishingly small deformations are obtained from classical linear

viscoelastic experiments.

{λ i , G i }

efﬁcient calculation in complex ﬂow geometries, but some stress equations have

been useful in particular classes of ﬂows.

Purely viscous constitutive equations, which account for some of the non-

linearity in shear but not for any of the history dependence, are commonly used

in process models when the deformation is such that the history dependence is

expected to be unimportant. The stress in an incompressible, purely viscous liquid

is of the form given in equation 2, but the viscosity is a function of one or more

invariant measures of the strength of the deformation rate tensor, [

Here, A (

where each term in the sum satisﬁes an equation of the form of equation 4, with a

spectrum of material parameters

266 PROCESSING, MODELING

Vol. 11

The stress constitutive equation can also be formulated as an integral over

the history of the deformation. The most common form used for simulations is

( t )

m ( t

−

t ) h (I C ,

II C ) C − 1 ( t

t ) dt

(6)

−∞

and is usually taken to be a sum of expo-

nentials. h(I C ,II C ) is a nonlinear “damping” function that depends on the ﬁrst and

second invariants of the Finger strain. Other strain measures may also be used,

and the popular K-BKZ model includes a small second term proportional to the

Cauchy strain measure C . The time–strain separability indicated by the product

of the functions m and h fails experimentally at very short times (9). As with the

differential equation forms, constitutive models of the general form of equation 6

follow from general continuum mechanics formulations, from transient-network

formulations, and from tube theories. There is sometimes a one-to-one equiva-

lence between a differential and integral equation formulation, but in general

only approximate equivalences can be developed. Most tube models are naturally

formulated as integral equations.

There are advantages and disadvantages to both differential and integral

constitutive equations. Differential equations adapt naturally to the numerical

methods commonly used for the solution of the momentum and energy equations,

which are in differential form, and most process modeling has been carried out

using differential stress constitutive equations. It is often important to include

the relaxation spectrum in process calculations, however, and each term in the

spectrum (eq. 5) requires an additional nonlinear partial differential equation

in the solution set, which greatly increases the magnitude of the computational

problem. The entire spectrum enters naturally in the integral formulation through

the memory function m ( t ), but it now becomes necessary to track the history of the

strain with great accuracy along material element paths on a computational grid

that cannot be laid out to ensure that particle paths pass through the spatial nodes.

Finally, there are some constitutive models that cannot be expressed in closed

form as differential or integral equations, but require the solution of a Fokker–

Plank equation (or an equivalent set of stochastic differential equations) for the

orientation distribution of chain segments in order to compute the stress (Ref-

erence 4, pp. 338 ff; Reference 10 and references therein). This technique may

become more useful as computing power increases, but to date it has been used

only for viscometric ﬂows and a few very simple non-viscometric geometries.

{λ i , G i }

Classiﬁcation

Classiﬁcations of viscoelastic ﬂows are useful for analysis, mainly in determining

conditions under which a viscoelastic constitutive equation that accounts for ﬂuid

memory can be replaced by a much simpler purely viscous equation.

Here, C − 1 is the Finger measure of strain and m ( t ) is a memory function that

can be determined from the linear viscoelastic response; m ( t ) can be expressed in

terms of the spectral parameters

Vol. 11

PROCESSING, MODELING 267

The broadest classiﬁcation scheme that has been found to be useful in model-

ing is based on the Deborah number (11–13), which is loosely deﬁned as the ratio

of the ﬂuid time scale to the time scale of the process. Small Deborah numbers

correspond to situations in which stress transients occur rapidly compared to the

total processing time; hence the “memory” of the ﬂuid is short and can be ne-

glected. Large Deborah number ﬂows take place over a time scale in which stress

transients cannot be neglected, and a viscoelastic description like that in equation

4 or 6 is essential. The signiﬁcance of this classiﬁcation is apparent in a linear

viscoelastic frequency sweep, where the Deborah number is the product of the

relaxation time

and the frequency

; for

λω

1 the storage mod-

ulus G dominates and the response is closer to that of an elastic solid. The ﬂuid

time scale is unambiguous for polymeric liquids described by constitutive equa-

tions such as equation 4; if the spectrum of relaxation times is required, as with

equation 5 or 6, an average or maximum relaxation time would be used, and there

is some ambiguity in the choice. There is frequently ambiguity in the choice of

the process time scale; the residence time is the obvious choice in a unidirectional

ﬂow, but the selection is not clear in more complex ﬂow ﬁelds.

The concept of strong and weak ﬂows (14) is one attempt to quantify the Debo-

rah number concept unambiguously. This classiﬁcation is based on the notion that

shearing ﬂows are an ineffective (weak) means of stretching polymer molecules,

whereas extensional ﬂows can be effective (strong) in elongating molecules. The

physical basis for the concept is rooted in the coil-stretch transition; a macro-

molecule modeled by a Rouse chain with a maximum relaxation time

λω

λ m will

2 .

(The “upper convected” Maxwell model is the continuum equivalent of a ﬂuid

made up of Rouse chains, and the stresses in an elongational ﬂow for this ﬂuid

model become unbounded when

when

λ m >

2 .) Let L be the velocity gradient, L

=∇

v ,

has a positive real part, in which case a Rouse chain will become fully extended

if the time in the ﬂow is large relative to a relaxation time; this is equivalent to

the condition

an eigenvalue of the matrix

λ m L

−

2 I . A ﬂow is strong if any eigenvalue

2 for pure extension. A processing ﬂow can be strong in some

regions and weak in others according to this classiﬁcation.

Classiﬁcation in terms of stress level has not been widely used, but stress

level is an important factor in process modeling. It is well known that process

behavior in conﬁned ﬂows changes dramatically when the wall stress in highly

entangled melts or concentrated solutions is comparable in magnitude to the shear

modulus, which is essentially equal to the plateau value of the storage modulus.

Flow instabilities like melt fracture (15) are observed, for example, and there is

an apparent failure of adhesion between the melt and the metal surface (15).

Low stresses are those for which

λ m >

denotes the magnitude

of the stress tensor and G p is the plateau modulus. This ratio is equivalent to

what is often called the recoverable shear and is proportional in channel ﬂow to a

dimensionless group known as the Weissenberg number, We

/ G p ≤

1, where

is a

characteristic relaxation time, v is the average velocity, and d is the diameter or

thickness of the channel. As discussed subsequently, the appropriate formulation

of wall boundary conditions for model equations is uncertain outside the low stress

regime.

= λ

v / d , where

1 the loss modulus G dominates

and the response is that of a viscous liquid, whereas for

be stretched out in an elongational ﬂow with stretch rate

and

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